Shakespeare creates the mood in The Tempest through three general techniques.
First, through the externalization of emotions and themes. The easiest example of this is the storm at the beginning that gives the play its name. It is a literal storm, but it is also a great emotional and political upset.
Second, through, well, special effects. Through magic, music, etc.
Third, through language. Look at the wonderful speeches by Prospero, Miranda, Caliban, etc. These use both images and metaphors to set the mood.